# How to summarize measurement variability for repeated measures?

I have a data set where different measures are taken repeatedly (3) times for multiple samples, like so:

   A B C
X  ...
Y  ...


So, for people X, Y, etc I have measures A, B, C and so on. Each of A, B, C are measured 3 times for each sample. That is, for each person, I have 3 measures of A (A1, A2, A3), 3 measures of B (B1, B2, B3), for each of the columns.

What I'd like to know is, how can I summarize the variability or the reliability of each column? So, over the entire data set, how reliable is the measurement for A?

Thanks for the help!

the sample variance of the measurements A1, A2, A3, for a person X is:

$$\sigma_X(A) = \frac{1}{2} \sum_{i=1}^{3} (A_{i,X} - \bar{A}_X)^2$$

where $\bar{A}_X$ is the arithmetic mean of the measurement A for person X. Now you can look at the $\sigma_X(A)$ values across all samples (people) in your study, and for example consider mean value $\bar{\sigma}(A)$ to quantify the reliability of measurements A.

$$\bar{\sigma}(A) = \frac{1}{N} \sum_X \sigma_X(A)$$

Another way of looking at it is if you consider each $A_i$, for $i=1,2,3$, across all samples. Define $\bar{A}_i$ as the mean value of the $A_i$ measurements across samples. This allows us to characterize the standard deviation $\sigma_i(A)$ of a particular measurement $A_i$ across samples.

$$\sigma_i(A) = \frac{1}{2} \sum_{X} (A_{i,X} - \bar{A}_i)^2$$

The average $\sigma_i(A)$ for all $i=1,2,3$ indicates the average variability of the measurement A.